Title: Sampling reduced density matrix to extract fine levels of entanglement spectrum

Abstract: Low-lying entanglement spectrum provides the quintessential fingerprint to identify the highly entangled quantum matter with topological and conformal field-theoretical properties. However, when the entangling region acquires long boundary with the environment, such as that between long coupled chains or in two or higher dimensions, there unfortunately exists no universal yet practical method to compute the entanglement spectra with affordable computational cost. Here we propose a new scheme to overcome such difficulty and successfully extract the low-lying fine entanglement spectrum (ES). We trace out the environment via quantum Monte Carlo simulation and diagonalize the reduced density matrix to gain the ES. We demonstrate the strength and reliability of our method through long coupled spin chains and answer its long-standing controversy. Our simulation results, with unprecedentedly large system sizes, establish the practical computation scheme of the entanglement spectrum with a huge freedom degree of environment.